As stated above, one can then study geodesics for this metric. It turns out that this (almost!) works as in the usual Riemannian setting: there is a set of well-behaved geodesics that admits a cotangent description (namely, they are projections of Hamiltonian orbits of the Legendre dual of \(f\) ). We restrict our attention to these. The geodesics that are left out are the so-called strict abnormals [1]}, [2]}, [3]}, [4]}: understanding them better is a central question in sub-Riemannian geometry (but that we will mostly ignore).
